Combining harmonic sampling with the worm algorithm to improve the efficiency of path integral Monte Carlo

We propose an improved Path Integral Monte Carlo (PIMC) algorithm called harmonic PIMC (H-PIMC) and its generalization, mixed PIMC (M-PIMC). PIMC is a powerful tool for studying quantum condensed phases. However, it often suffers from a low acceptance ratio for solids and dense confined liquids. We develop two sampling schemes especially suited for such problems by dividing the potential into its harmonic and anharmonic contributions. In H-PIMC we generate the imaginary time paths for the harmonic part of the potential exactly and accept or reject it based on the anharmonic part. In M-PIMC we restrict the harmonic sampling to the vicinity of local minimum and use standard PIMC otherwise, in order to optimize efficiency. We benchmark H-PIMC on systems with increasing anharmonicity, improving the acceptance ratio and lowering the autocorrelation time. For weakly to moderately anharmonic systems, at βℏω=16, H-PIMC improves the acceptance ratio by a factor of 6-16 and reduces the autocorrelation time by a factor of 7-30. We also find that the method requires a smaller number of imaginary time slices for convergence, which leads to another two- to threefold acceleration. For strongly anharmonic systems, M-PIMC converges with a similar number of imaginary time slices as standard PIMC but allows the optimization of the autocorrelation time. We extend M-PIMC to periodic systems and apply it to a sinusoidal potential. Finally, we combine H- and M-PIMC with the worm algorithm, allowing us to obtain similar efficiency gains for systems of indistinguishable particles.